Bachmann showed (in particular)
that if \(A\)  is a compact object in \(SH(k)\) 
such that \(M(A)=0\)  in \(DM(k)\)  and for every \(\sigma \) 
in the space \(\operatorname{Sper}(k)\)  of orderings of \(k\) ,
\(M_{\sigma }[1/2](A)=0\)  in \(D(\text{\bf Z}[1/2])\) , then \(A=0\) 
[1]}.
When \(\sigma \) 
comes from an embedding of \(k\)  into \(\text{\bf R}\) , \(M_{\sigma }[1/2](A)\) 
is the complex that computes the singular homology of the corresponding
real realization of \(A\) , \(H_*(A(\text{\bf R}),\text{\bf Z}[1/2])\) 
[1]}. It remains to show
that we only need to consider orderings that
come from real embeddings of \(k\) .
We use the following property of the space
\(\operatorname{Sper}(k)\)  of orderings of \(k\)  [3]}.
The topology on \(\operatorname{Sper}(k)\)  is defined by taking the sets
\(\lbrace \sigma : a>_{\sigma }0\rbrace \)  for \(a\in k\)  as a sub-basis for the topology.
This makes \(\operatorname{Sper}(k)\)  into a compact Hausdorff totally
disconnected space.
Lemma 8.2 Let \(k\)  be a finitely generated field of characteristic zero.
Then the set of archimedean orderings of \(k\)  is dense
in the topological space \(\operatorname{Sper}(k)\)  of orderings of \(k\) ,
and every archimedean ordering comes from an embedding
of \(k\)  into \(\text{\bf R}\) .

Given that, we are done if we can show that the support
in \(X:=\operatorname{Sper}(k)\)  of a compact object in \(SH(k)\)  is open as well as closed.
This is related to the general fact that for a tensor triangulated
category \(K\) , the support of an object of \(K\)  in the Balmer spectrum
\(\operatorname{Spc}(K)\)  is closed and its complement is quasi-compact
[4]}. However,
we will argue more directly.
We use that the functors \(M_{\sigma }\) 
come from a functor from \(SH(k)\)  to the derived category of sheaves
\(D(X,\text{\bf Z}[1/2])\) ,
which takes compact objects to compact objects. Indeed, by
[1]}, the functor
from \(SH(k)\)  to Witt motives \(DM_W(k,\text{\bf Z}[1/2])\)  is monoidal;
so it takes rigid objects
to rigid objects, and the rigid objects
coincide with the compact objects in these categories. (Some people
say “strongly dualizable” rather than “rigid”.)
Furthermore,
\(DM_W(k,\text{\bf Z}[1/2])\)  is equivalent to \(D(X,\text{\bf Z}[1/2])\) 
[1]}.
A compact object in \(D(X,\text{\bf Z}[1/2])\)  is a perfect complex;
that is, it is locally
isomorphic to a bounded complex of finitely generated
projective \(\text{\bf Z}[1/2]\) -modules.
(Indeed, since \(X\)  is compact, Hausdorff,
and totally disconnected, every open subset of \(X\)  is a union
of clopen subsets (or equivalently, quasi-compact open subsets).
It follows that every compact object
in \(D(X,\text{\bf Z}[1/2])\)  is a summand in \(D(X,\text{\bf Z}[1/2])\)  of a bounded
complex of sheaves which are finite direct sums of sheaves
of the form \(j_!(\text{\bf Z}[1/2]_U)\) , with \(j\colon U\hookrightarrow X\) 
the inclusion of a quasi-compact open subset [7]}.
Clearly such a summand is a perfect complex of \(\text{\bf Z}[1/2]\) -modules
on \(X\) .)
Because sections of the sheaf \(\text{\bf Z}[1/2]\)  on \(X=\operatorname{Sper}(k)\)  are locally
constant, the support of a perfect complex on \(X\) 
is open as well as closed.
